A generalized Bach tensor $B_{ij}^t$ with parameter $t\in \mathbb{R}$ is introduced. A Riemannian manifold $(M^n,g)$ is called $B^t$-flat if its generalised Bach tensor $B_{ij}^t\equiv 0$ for some parameter t. In this paper, we first study the rigidity of closed $B^t$-flat Riemannian manifolds with positive constant scalar curvature. When the dimension $n=4$, we prove that all $B^t$-flat manifolds that satisfy a point-wise inequality must be of positive constant sectional curvature. Similar rigidity results are also obtained in terms of the Yamabe invariant. Moreover, using the curvature estimates for the general dimension $n\ge 4$, we obtain an integral inequality for $B^t$-flat manifolds, and prove that the equality occurs if and only if these manifolds are of positive constant sectional curvature. In addition, we also obtain similar rigidity results for complete $B^t$-flat manifolds.

广义巴赫张量 ${乙}_{\mathrm{一世}j}^{吨}$ 带参数 $吨\in \mathbb{电阻}$介绍。黎曼流形$({米}^n,G)$ 叫做 ${乙}^{吨}$-flat 如果它的广义巴赫张量 ${乙}_{\mathrm{一世}j}^{吨}\equiv 0$对于某些参数t。在本文中，我们首先研究封闭的刚性${乙}^{吨}$-具有正常数标量曲率的平坦黎曼流形。当维度$n=4$，我们证明所有 ${乙}^{吨}$-满足逐点不等式的平坦流形必须具有正的恒定截面曲率。在 Yamabe 不变量方面也获得了类似的刚性结果。此外，使用一般维度的曲率估计$n\ge 4$，我们得到一个积分不等式 ${乙}^{吨}$-扁平流形，并证明当且仅当这些流形具有正的恒定截面曲率时，等式才会发生。此外，我们还获得了完全相似的刚性结果${乙}^{吨}$-扁平歧管。